Calculus BC Review Sheet

Q: What continuity test is used at a point a?

A function is continuous at “a” when the left-hand limit, the right-hand limit, and the function value match at that point.

Q: When can L’Hôpital’s Rule be applied?

It applies for indeterminate forms where both the function limits approach 0 or both approach ±∞; then the limit of the functions equals the limit of their derivatives, if defined.

Q: How do you estimate integrals on an interval using Riemann sums?

Partition the interval into subintervals with width Δx, compute heights from function values at chosen sample points, and sum Δx times those heights; the sheet also notes variations based on over/under estimation.

Sep 20, 2026

Uploaded by Ava Thompson

Calculus BC Review Sheet consolidates core limits and continuity criteria, derivative definition, and key differentiability and graph-analysis workflows. It includes differentiation rules (constant, power, product, quotient, chain), special derivative families (trigonometric, inverse trigonometric, logarithmic and exponential, inverse functions), and major theorems such as Mean Value, Rolle’s, Intermediate Value, and Extreme Value. The sheet further covers integral foundations, estimating integrals via Riemann sums and trapezoidal rule, exact area between curves, volume and arc length formulas, and integral techniques like substitution and basic integration patterns.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Limits and Continuity

L’Hôpital’s Rule and Discontinuity Types

Derivatives: Definition and Rules

Derivative Applications and Theorems

Graph Analysis: Critical Points and Tests

Speed and Motion Concepts

Fundamental Theorem of Calculus

Estimating Integrals: Riemann Sums and Trapezoidal Rule

Area, Volume, and Arc Length

Integral Rules and U-Substitution

Calculus BC Review Sheet | LIMITS
lim 􀝂(􀝔) = 􀜮 exists if and only 􀯫→􀯔 | if lim
􀯫→􀯔 − | 􀝂 (􀝔) = lim 􀝂 (􀝔) = 􀜮􀯫→􀯔+ | | CONTINUITY
A function is continuous at “a” if
􀯫lm􀯔− 􀝂 (􀝔) = 􀯫lm􀯔+ 􀝂 (􀝔) = 􀝂 (􀜽) | | | | | | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | L’HOPITAL’S RULE
If lim 􀝂(􀝔) = 0 and lim 􀝃 (􀝔) = 0
􀯫→􀯖 􀯫→􀯖
or
lim 􀝂(􀝔) = ±∞ and lim 􀝃(􀝔) = ±∞􀯫→􀯖 􀯫→􀯖
then lim 􀯙 (􀯫) = lim 􀯙′(􀯫) = 􀜮 .
􀯫→􀯖 􀯚 (􀯫) 􀯫→􀯖 􀯚′(􀯫) | | | | | | | | | | | | | | | TYPES OF DISCONTINUITY

Point or Removable | |
Infinte or Asymptote | | | | | DEFINITION OF THE DERIVATIVE
􀝂 ′(􀝔) = lim 􀝂 (􀝔 + ℎ) − 􀝂(􀝔)
ℎ→0 ℎ | | | | NOT DIFFERENTIABLE | | | | | | | | | | | | | | | | | | | | | | Corner Cusp | | | | | | | | | | | | | | | | | | **If a function has a derivative in its domain, then the function is DIFFERENTIABLE in its given domain | | | | | | | | | | | | | | | Discontinuity Vertical Tangent | | | | | | | DERIVATIVE RULES CONSTANT: 􀯗􀯗􀯫 [􀜿] = 0
CONSTANT MULTIPLIER: 􀯗􀯗􀯫 [􀜿 􀝂(􀝔)] = 􀜿 ∙ 􀝂′(􀝔)
POWER RULE: 􀯗􀯗􀯫 [􀝔􀯡 ] = 􀝊􀝔􀯡−1 PRODUCT RULE
􀝀􀝀􀝔 [􀝂 (􀝔) ∙ 􀝃 (􀝔)] = 􀝂 (􀝔)􀝃 ′(􀝔) + 􀝃 (􀝔)􀝂 ′(􀝔)
QUOTIENT RULE
􀝀􀝀􀝔 [􀝂􀝃((􀝔􀝔))] = 􀝃~~ ~~(􀝔)􀝂~~ ~~′(􀝔[􀝃)()􀝂]2(􀝔)􀝃~~ ~~′(􀝔)
" Lo(dHi)– Hi(dLo) "
Lo2
CHAIN RULE
􀝀 ′ ′􀝀􀝔 [􀝂 (􀝃 (􀝔))] = 􀝂 (􀝃 (􀝔)) ∙ 􀝃 (􀝔)
􀝀􀝀􀝔 [􀝂(9 − 3􀝔)] = 􀝂′(9 − 3􀝔) ∙ (−3) | | | | TRIG DERIVATIVES
􀝀􀝀􀝔 [sin 􀝑] = cos 􀝑 ∙ 􀝀􀝀􀝑􀝔􀝀􀝀􀝔 [cos 􀝑] = − sin 􀝑 ∙ 􀝀􀝀􀝑􀝔􀝀􀝀􀝔 [tan 􀝑] = sec 2 􀝑 ∙ 􀝀􀝀􀝑􀝔􀝀􀝀􀝔 [cot 􀝑] = − csc2 􀝑 ∙ 􀝀􀝀􀝑􀝔􀝀􀝀􀝔 [sec 􀝑] = sec 􀝑 tan 􀝑 ∙ 􀝀􀝀􀝑􀝔􀝀􀝀􀝔 [csc 􀝑] = − csc 􀝑 cot 􀝑 ∙ 􀝀􀝀􀝑􀝔
INVERSE TRIG DERIVATIVES
􀝀 [sin−1 􀝑] = 􀝑′ 􀝀􀝔 √1 − 􀝑2
􀝀 􀝑′ [tan−1 􀝑] =
􀝀􀝔 1 + 􀝑2
LOG, EXPONENTIAL DERIVATIVES
􀝀􀝀􀝔 [􀝁 􀯚 (􀯫)] = 􀝁 􀯚 (􀯫) ∙ 􀝃 ′(􀝔)
􀝀 1 ′[ln 􀝃 (􀝔)] = ∙ 􀝃 (􀝔)
􀝀􀝔 􀝃 (􀝔)
DERIVATIVE OF INVERSES
􀝀􀝀􀝔 [􀝂 −1 (􀝔)] = 􀝂~~ ~~′~~ ~~[􀝂~~ ~~1~~ ~~(􀝔)] | | | | | | | THEOREMS | | | | | | | | | | | MEAN VALUE THM
Iff is continuous on [􀜽, 􀜾] and differentiable on (􀜽, 􀜾), then there exists a c on the interval (􀜽, 􀜾) such that
􀝂 ′(􀜿) = 􀝂 (􀜾) − 􀝂 (􀜽)
􀜾 − 􀜽
| | | ROLLE’S THM
Iff is continuous on [􀜽, 􀜾], differentiable on (􀜽, 􀜾), and 􀝂(􀜽) =􀝂 (􀜾), then there exists a c on the interval (􀜽, 􀜾) such that
􀝂 ′(􀜿) = 􀝂~~ ~~(􀜾􀜾)~~ ~~−~~ ~~􀝂􀜽(􀜽) = 0 | | INTERMEDIATE VALUE THM
Iff is continuous on [􀜽, 􀜾] and there is a value of k between 􀝂(􀜽) and 􀝂(􀜾), then there exists at least on value of c on (􀜽, 􀜾)
such that 􀝂(􀜿) = 􀝇 . | | | EXTREME VALUE THM
Iff is continuous on [􀜽, 􀜾], then f has an absolute maximum and an absolute minimum value at 􀝔 = 􀜽,􀝔 = 􀜾 , or when
􀝂 ′(􀝔) = 0 or 􀝂′(􀝔) is undefined. | | | GRAPH ANALYSIS | | | | | | | | | | | If 􀝂′(􀝔)… Then 􀝂(􀝔)… … is 0 or undefined at 􀝔 = 􀜽 has critical points and has a
potential relative extrema at 􀝔 = 􀜽 … is positive … is increasing … is negative … is decreasing … changes from positive to negative …􀝂 (􀜽) is a relative maximum … changes from negative to positive …􀝂 (􀜽) is a relative minimum | | | | If 􀝂 ′′(􀝔)… Then 􀝂(􀝔)… … is 0 or undefined at 􀝔 = 􀜽 …has a potential point of inflection … is positive … is concave up … in negative … is concave down … changes from signs …(􀜽, 􀝂 (􀜽)) is a point of inflection
2nd DERIVATIVE TEST
Let f be a function such that the second derivative of f exists on an open interval containing c.
1) If 􀝂 ′(􀜿) = 0 and 􀝂 ′′(􀜿) > 0, then 􀝂 (􀜿) is a relative minimum.
2) If 􀝂′(􀜿) = 0 and 􀝂 ′′(􀜿) < 0, then 􀝂 (􀜿) is a relative maximum | | | | | | | SPEED
Speed = |􀝒(􀝐)|
SPEED Increasing
Velocity and Acceleration have the same sign
SPEED Decreasing
Velocity and Acceleration have opposite sign | | Displacement:
The change in distance from where the object begins and where it ends.
On the interval [a, b], where 􀝌(􀝐) is the position function…
I. 􀝌 (􀜾) − 􀝌 (􀜽)
II. ∫􀯕􀯔 􀝒 (􀝐) 􀝀􀝐 | | Total Distance:
The sum of all distances moved in any direction.
􀯕
∫ |􀝒

FAQs

What continuity test is used at a point a?

A function is continuous at “a” when the left-hand limit, the right-hand limit, and the function value match at that point.

When can L’Hôpital’s Rule be applied?

It applies for indeterminate forms where both the function limits approach 0 or both approach ±∞; then the limit of the functions equals the limit of their derivatives, if defined.

How do you estimate integrals on an interval using Riemann sums?

Partition the interval into subintervals with width Δx, compute heights from function values at chosen sample points, and sum Δx times those heights; the sheet also notes variations based on over/under estimation.

Sep 20, 2026

Uploaded by Ava Thompson